Let the system
be the input to the sampler,
be the sampler output,
be the input to the sampler, and
be the sampler output.

Step3:

Consider each sample output to be a source node.

…… (1)

...… (2)

.….. (3)

Where,

Step4:

Take the starred transform for Equations (1), (2), and (3).

…… (4)

…… (5)

…… (6)

Substitute Equation (5) in Equation (4) and find:

Substitute the
in Equation (6):

Thus, the transfer function of this system is.

b)

Find the expression
as a function of the input and output of the transfer function of
the system as shown in Figure 1.

Figure 3: Block diagram of the given system.

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Step 4 of 18

Step1:

Construct the original signal flow graph as shown in Figure
(4).

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Step 5 of 18

Figure 4: Signal flow graph.

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Step 6 of 18

Step2:

Assign a variable to each sampler input.

Let the system
be the input to the sampler,
be the sampler output,
be the input to the sampler, and
be the sampler output.

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Step 7 of 18

Step3:

Consider each sample output to be a source node.

…… (1)

…… (2)

…… (3)

Where,

Step4:

Take the starred transform for Equations (1), (2), and (3).

…… (4)

…… (5)

…… (6)

Substitute the
in Equation (5):

…… (7)

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Alexander Mussa

G*H* should be GH* with a bar over the top

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Step 8 of 18

Substitute
and
in Equation (6):

Thus, the transfer function of the system is:
.

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Alexander Mussa

You forgot to multiply by G* in the second step and thus the answer should have G(z) in the numerator

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Step 9 of 18

c)

To find the express
as a function of the input and output of the transfer function as
shown in Figure (5)

Figure (5): Block diagram of the given system

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Step 10 of 18

Step1:

Construct the original signal flow graph as shown in Figure
(6)

Figure (6): Signal flow graph

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Step 11 of 18

Step2:

Assign a variable to each sampler input.

Let the system
be the input to the sampler and
be the sampler output.

Step3:

Consider each sample output to be a source node:

…... (1)

…… (2)

Where,

.

Step4:

Take the starred transform of Equation (1)

…… (3)

Take the starred transform of Equation (2):

…… (4)

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Step 12 of 18

Substitute
in Equation (4).

Thus, the transfer function of this system is
.

d)

To find the express
as a function of the input and output of the transfer function as
shown in Figure (7)

Figure (7): Block diagram of the given system

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Step 13 of 18

Step1:

Construct the original signal flow graph as shown in Figure
(8)

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Step 14 of 18

Figure (8): Signal flow graph

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Step 15 of 18

Step2:

Assign a variable to each sampler input.

Let the system
be the input to the sampler and
be the sampler output.

Step3:

Consider each sample output to be a source node:

…… (1)

…… (2)

Where,

Step4:

Take the starred transform of Equations (1) and (2):

…… (3)

…… (4)

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Step 16 of 18

Substitute
in Equation (4).

Thus, the transfer function of the system is
.

e)

To find the express C(z) as a function of the
input and output of the transfer function as shown in Figure
(9)

Figure (9): Block diagram of the given system

Step1:

Construct the original signal flow graph as shown in Figure
(10)

Figure (10): Signal flow graph

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Anonymous

In step 16, C was forgotten when simplified to C(1+G+HG). There should be no 'C(z)' value in the final solution. Therefore the actual transfer function should be C/R = G/(1+G+HG) in the z-transform notation.

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Step 17 of 18

Step2:

Assign a variable to each sampler input.

Let the system
be the input to the sampler,
be the sampler output,
be the input to the sampler, and
be the sampler output.

Step3:

Consider each sample output and input to be a source node:

…… (1)

…… (2)

Where,

Step4:

Take starred transform for Equations (1) and (2):

…… (3)

…… (4)

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Step 18 of 18

Substitute
in Equation (4).

Thus, the transfer function of the system is
.

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Anonymous

Step 18 (like step 16) forgot to simplify correctly to include the single C* there was.... so the combination should be C*(1+G*H*G*) and follow through to the end of the problem.

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